Enhanced shape recovery in advection--diffusion problems via a novel ADMM-based CCBM optimization
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915870344740864 |
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| author | Cherrat, Elmehdi Afraites, Lekbir Rabago, Julius Fergy Tiongson |
| author_facet | Cherrat, Elmehdi Afraites, Lekbir Rabago, Julius Fergy Tiongson |
| contents | This work proposes a novel shape optimization framework for geometric inverse problems governed by the advection--diffusion equation, based on the coupled complex boundary method (CCBM). Building on recent developments [Afr22, Rab23, Rab25, RAN25, RN24], we aim to recover the shape of an unknown inclusion via shape optimization driven by a cost functional constructed from the imaginary part of the complex-valued state variable over the entire domain. We rigorously derive the associated shape derivative in variational form and provide explicit expressions for the gradient and second-order information. Optimization is carried out using a Sobolev gradient method within a finite element framework. To address difficulties in reconstructing obstacles with concave boundaries, particularly under measurement noise and the combined effects of advection and diffusion, we introduce a state-of-the-art numerical scheme inspired by the Alternating Direction Method of Multipliers (ADMM). In addition to implementing this non-conventional approach, we demonstrate how the adjoint method can be efficiently applied and utilize partial gradients todevelop a more efficient CCBM-ADMM scheme. The accuracy and robustness of the proposed computational approach are validated through various numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enhanced shape recovery in advection--diffusion problems via a novel ADMM-based CCBM optimization Cherrat, Elmehdi Afraites, Lekbir Rabago, Julius Fergy Tiongson Numerical Analysis Optimization and Control 49Q10, 49K20, 65K10 This work proposes a novel shape optimization framework for geometric inverse problems governed by the advection--diffusion equation, based on the coupled complex boundary method (CCBM). Building on recent developments [Afr22, Rab23, Rab25, RAN25, RN24], we aim to recover the shape of an unknown inclusion via shape optimization driven by a cost functional constructed from the imaginary part of the complex-valued state variable over the entire domain. We rigorously derive the associated shape derivative in variational form and provide explicit expressions for the gradient and second-order information. Optimization is carried out using a Sobolev gradient method within a finite element framework. To address difficulties in reconstructing obstacles with concave boundaries, particularly under measurement noise and the combined effects of advection and diffusion, we introduce a state-of-the-art numerical scheme inspired by the Alternating Direction Method of Multipliers (ADMM). In addition to implementing this non-conventional approach, we demonstrate how the adjoint method can be efficiently applied and utilize partial gradients todevelop a more efficient CCBM-ADMM scheme. The accuracy and robustness of the proposed computational approach are validated through various numerical experiments. |
| title | Enhanced shape recovery in advection--diffusion problems via a novel ADMM-based CCBM optimization |
| topic | Numerical Analysis Optimization and Control 49Q10, 49K20, 65K10 |
| url | https://arxiv.org/abs/2508.16898 |